Some identities on the higher order twisted q Euler numbers and polynomials with weight α

R E S E A R C H

Open Access

Some identities on the higher-order-twisted

q

-Euler numbers and polynomials with weight

a

Hui Young Lee

, Nam Soon Jung, Jung Yoog Kang and Cheon Seoung Ryoo

* Correspondence: [email protected] Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract

In this article, we introduce some properties of higher-order-twistedq-Euler numbers and polynomials with weighta, and we observe some properties of higher-order-twistedq-Euler numbers and polynomials with weighta for several cases. In

particular, by using the the fermionicp-adicq-integral onℤ_p, we give a new concept of twistedq-Euler numbers and polynomials with weighta.

2000 Mathematics Subject Classification: 11B68; 11S40; 11S80.

Keywords:Euler numbers and polynomials,q-Euler numbers and polynomials, higher-order-twistedq-Euler numbers and polynomials with weightα

1. Introduction

Letpbe a fixed odd prime. Throughout this articleℤp,Qp,ℂ, andℂp, will, respectively,

denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, the complex number field, and the completion of algebraic closure ofQp. LetN be the set of natural numbers andℤ+=N∪{0}. Letνpbe the normalized exponential valuation of ℂp withpp=p−νp

(p)_=p−1

(see [1-14]). When one speaks of q-extension, q can be regarded as an indeterminate, a complex numberq Îℂ, orp-adic numberqÎ ℂp; it

is always clear from context. Ifq Îℂ, we assume |q| < 1. IfqÎ ℂp, then we assume |

1 -q|p< 1 (see [1-14]).

In this article, we use the notation ofq-number as follows (see [1-14]):

[x]q=

1−qx

1−q.

Note that limq®1[x]q=xfor any xwith |x|p≤1 in thep-adic case.

LetC(ℤp) be the space of continuous functions onℤp. ForfÎC(ℤp), Kim defined the

fermionicp-adicq-integral onℤpas follows (see [6,7]):

I₋q(f) =

Zp

f(x)dμ₋q(x) = lim N→∞

1 [pN_]

−q pN₋₁

x=0

f(x)(−q)x,

= lim

N→∞

[2]_q 1 +qpN

pN₋₁

x=0

f(x)(−q)x.

(1)

From (1), we note that

qI₋q(f1) +I−q(f) = [2]qf(0), wheref1(x) =f(x+ 1).

It is well known that the ordinary Euler polynomials are defined by

2

et_{+ 1}e

xt_=eE(x)t₌

∞

n=0

En(x)

tn n!,

with the usual convention of replacingEn(x) byEn(x).

In the special case,x= 0,En(0) =Enare called thenth Euler numbers (see [1-14]).

By (2), we get the following recurrence relation as follows:

E0= 1, and (E+ 1)n+E=

2, ifn= 0,

0, ifn>0. (2)

Recently, (h, q)-Euler numbers are defined by

E(_0,h_q)= 2

1 +qh, andq h_(qE(h)

q + 1)n+E(qh)=

2, ifn= 0, 0, ifn>0,

with the usual convention about replacing

E(qh) n

byE(nh,q)(see [1-16]).

Note thatlimq→1E(nh,q)=En.

LetTp=∪N≥1CpN = lim_N→∞C_pN, whereC_pN ={w|wp N

= 1}is the cyclic group of order

pN. ForwÎTp, we denote byjw:ℤp® ℂpthe locally constant functionx↦wx.

ForaÎNandwÎTp, the twistedq-Euler numbers with weightaare also defined by

˜

E(_0,α_q)_,w= [2]q

wq+ 1, andwq

qαE˜(qα,w)+ 1 n

+E˜(nα,q),w=

[2]_q, ifn= 0, 0, ifn>0,

with the usual convention about replacingE˜(qα,w) n

byE˜(nα,q),w(see [2,5]).

The main purpose of this article is to present a systemic study of some families of higher-order-twistedq-Euler numbers and polynomials with weighta. In Section 2, we investigate higher-order-twistedq-Euler numbers and polynomials with weightaand establish interest-ing properties. In Sections 3, 4, and 5, we observe some properties for special cases.

2. Higher-order-twisted q-Euler numbers and polynomials with weight a For hÎ ℤ, a, kÎ N, wÎ Tp and nÎ ℤ+, let us consider the expansion of higher-order-twistedq-Euler polynomials with weight aas follows:

˜

E(nα,q),w(h,k|x)

=

Zp

· · ·

Zp

w

k−times

k

i=1xi

_k

i=1

xi+x n

qα

qx1(h−1)+···+xk(h−k)_dμ

−q(x1)· · ·dμ−q(xk). (3)

From (1) and (3), we note that

˜

E(nα,q),w(h,k|x) =

[2]k q

(1−qα)n

n

l=0

n l

(−1)l q

αlx

In the special case, x = 0 E˜(nα,q),w(h,k|0) =E˜

(α)

n,q,w(h,k)are called the higher-order-twisted q-Euler numbers with weight a.

By (3), we get

˜

En(α,q),w(h,k) = (qα−1)E˜(nα+1,)q,w(h−α,k) +E˜

(α)

n,q,w(h−α,k). (5)

From (5) and mathematical induction, we get the following theorem. Theorem 1. Fora, kÎ NandnÎℤ+, we have

From (3), (4), and above property, we have

˜

From (3), we can derive the following equation.

By (3), (4), (5), and (6), we see that

i

j=0

(qα−1)j

i j

˜

E(_nα₋)_1+j,q,w(h−α,k) =

i−1

j=0

(qα−1)j

i−1

j

˜

E(_nα₋)_i+j,q,w(h,k).

Therefore, we obtain the following theorem. Theorem 2. Fora, kÎ Nandn, iÎℤ+, we have

i

j=0

i j

(qα−1)jE˜(_nα₋)_i+j,q,w(h−α,k) =

i−1

j=0

(qα−1)j

i−1

j

˜

E(_nα₋)_i+j,q,w(h,k).

By simple calculation, we easily see that m

j=0

m j

(qα−1)jE˜(_j,α_q)_,w(0,k) = [2]

k q

(1 +wqαm_{)(1 +wq}αm−1_{)· · ·(1 +wq}αm−k+1₎.

3. Polynomials E˜(nα,q),w(0,k|x)

We now consider the polynomialsE˜(nα,q),w(0,k|x)(inqx) by

˜

E(nα,q),w(0,k|x)

=

Zp

· · ·

Zp

k−times

wx1+···+xk

x+

k

i=1

xi

n

qα

q−kj=1jxj_dμ

−q(x1)· · ·dμ−q(xk). (8)

By (8) and (4), we get

(qα−1)nE˜n(α,q),w(0,k|x) = [2]kq n

l=0

n l

qαlx(−1)n−1 1

(1 +wqαl_{)· · ·(1 +wq}αl−k+1₎. (9)

From (8) and (9), we can derive the following equation.

Zp

· · ·

Zp

wx1+···+xk_q k

j=1(αn−j)xj+αnx_dμ

−q(x1)· · ·dμ−q(xk)

=

n

j=0

n j

[α]jq(q−1)jE˜

(α)

j,q,w(0,k|x),

and

Zp

· · ·

Zp

wx1+···+xk_qkj=1(αn−j)xj+αnx_dμ

−q(x1)· · ·dμ−q(xk)

= [2]

k qqαnx

(1 +wqαn_{)· · ·(1 +wq}αn−k+1₎.

(10)

Therefore, by (9) and (10), we obtain the following theorem. Theorem 3. ForaÎNandn, kÎℤ+, we have

˜

E(nα,q),w(0,k|x) =

[2]k_q [α]n

q(1−q)n n

l=0

n l

(−1)lqαlx 1

(−wqαl−k+1_:q)

k

and n

l=0

n l

[α]l_q(q−1)lE˜_l(_,α_q)_,w(0,k|x) = q

αnx_[2]k q

(−wqαn−k+1_:q)

k

,

where (a: q)0= 1 and (a:q)k= (1 -a)(1 -aq) ... (1 -aqk-1).

Let dÎNwithd≡ 1 (mod 2). Then we have

Zp · · ·

Zp

wx1+···+xk

⎡ ⎣x+

k

j=1 xj

⎤ ⎦

n

qα

q−kj=1jxj_dμ−

q(x1)· · ·dμ−q(xk)

= [d]

n qα [d]k

−q d−1

a1,···,ak=0

wa1+···+ak_q−

k

j=2(j−1)aj₍−₁₎kj=1aj×

Zp · · ·

Zp

wd(x1+···+xk)

⎡ ⎣x+

k j=1aj

d +

k

j=1 xj

⎤ ⎦

n

qαd

q−dkj=1jxj_dμ−

qd(x₁)· · ·dμ−_qd(x_k)

(11)

Thus, by (11), we obtain the following theorem. Theorem 4. FordÎ Nwithd≡1 (mod 2), we have

˜

E(nα,q),w(0,k|x)

= [d]

n qα

[d]k₋q d−1

a1,···,ak=0

(−w)a1+···+ak_q−kj=2(j−1)aj_E˜(α)

n,qd_,wd

0,k|x+a1+· · ·+ak d

.

Moreover, ˜

E(nα,q),w(0,k|dx)

= [d]

n qα

[d]k

−q d−1

a1,···,ak=0

(−w)a1+···+ak_q−kj=2(j−1)aj_E˜(α)

n,qd_,wd

0,k|x+a1+· · ·+ak

d

.

By (8), we get

˜

E(nα,q),w(0,k|x= n

l=0

n l

[x]n_q−αlqαlxE˜(_l,α_q)_,w(0,k)

=[x]_qα +qαxE˜(qα,w)(0,k) n

,

whereE˜(nα,q),w(0,k|0) =E˜(nα,q),w(0,k). Thus, we note that

˜

E(nα,q),w(0,k|x+y= n

l=0

n l

[y]nqα−lqαlyE˜_l(_,α_q)_,w(0,k|x)

=[y]_qα+qαyE˜(qα,w)(0,k|x) n

.

4. Polynomials E˜(nα,q),w(h, 1|x)

Let us define polynomials E˜(nα,q),w(h, 1|x)as follows:

˜

E(nα,q),w(h, 1|x) =

Zp

wx1_[x+x

From (12), we have

˜

E(nα,q),w(h, 1|x) =

[2]_q (1−qα)n

n

l=0

n l

(−1)lqαlx 1

(1 +wqαl+h₎.

By the calculation of the fermionic p-adicq-integral onℤp, we see that

qαx

Zp

wx1_[x+x

1]nqαqx1(h−1)dμ−q(x1)

= (qα−1)

Zp

wx1_[x+x

1]nqα+1qx1(h−α−1)dμ₋q(x1) +

Zp

wx1_[x+x

1]nqαqx1(h−α−1)dμ₋q(x1). (13)

Thus, by (13), we obtain the following theorem. Theorem 5. ForaÎNandhÎℤ, we have

qαxE˜(nα,q),w(h, 1|x) = (qα−1)E˜

(α)

n+1,q,w(h−α, 1|x) +E˜

(α)

n,q,w(h−α, 1|x). It is easy to show that

˜

E(nα,q),w(h, 1|x) =

Zp

wx1_[x+x

1]nqαqx1(h−1)dμ−q(x1) =

n

l=0

n l

[x]n_qα−1qαlx

Zp

wx1_[x

1]lqαqx1(h−1)dμ−q(x1)

=

n

l=0

n l

[x]n_qα−1qαlxE˜(_l,α_q)_,w(h, 1)

=

qαxE˜q(,αw)(h, 1) + [x]qα n

, forn≥1,

(14)

with the usual convention about replacing(E˜(qα,w)(h, 1))nbyE˜(nα,q),w(h, 1). FromqI-q(f1) +I-q(f) = [2]qf(0), we have

wqh

Zp

wx1_[x+x

1+ 1]nqαqx1(h−1)dμ−q(x1) +

Zp

wx1_[x+x

1]nqαqx1(h−1)dμ−q(x1) = [2]q[x]nqα.

(15)

By (13) and (15), we get

wqhE˜(nα,q),w(h, 1|x+ 1) +E˜

(α)

n,q,w(h, 1|x) = [2]q[x]nqα. (16) Forx= 0 in (16), we have

wqhE˜(nα,q),w(h, 1|1) +E˜

(α)

n,q,w(h, 1) =

[2]_q, ifn= 0,

0, ifn>0. (17)

Therefore, by (14) and (17), we obtain the following theorem. Theorem 6. ForhÎ ℤand nÎ ℤ+, we have

wqh(qαE˜(qα,w)(h, 1) + 1)nE˜

(α)

n,q,w(h, 1) =

[2]_q, ifn= 0, 0, ifn>0,

with the usual convention about replacing(E˜q(,αw)(h, 1))nbyE˜

(α)

From the fermionic p-adicq-integral onℤp, we easily get

˜

E(_0,α_q)_,w(h, 1) =

Zp

wx1_qx1(h−1)_dμ

−q(x1) = [2]_q [2]_wqh

.

By (12), we see that

˜

E(_nα_,q)−1_,w−1(h, 1|1−x) =

Zp

w−1[1−x+x1]nq−αq−x1(h−1)dμ−q−1(x₁)

= (−1)nwqαn+h−1 [2]q

(1−qα)n

n

l=0

n l

(−1)lqαlx 1

1 +wqαl+h

= (−1)nwqαn+h−1E˜n(α,q),w(h, 1|x)

(18)

Therefore, by (18), we obtain the following theorem. Theorem 7. ForaÎN,hÎℤandnÎℤ+, we have

˜

E(_nα_,q)−1_,w−1(h, 1|1−x) = (−1)nwqαn+h−1E˜

(α)

n,q,w(h, 1|x). In particular, forx= 1, we get

˜

E(nα,q),w(h, 1)

= (−1)nwqαn+h−1E˜n(α,q),w(h, 1|1)

= (−1)n+1qαn−1E˜n(α,q),w(h, 1) ifn≥1. Let dÎNwithd≡ 1 (mod 2). Then we have

Zp

wx1_qx1(h−1)_[x+x

1]nqαdμ−q(x1)

= [d]

n qα

[d]_−q

d−1

a=0

waqha(−1)a

Zp

wdx1

_x+a

d +x1

n

qαdq

x1(h−1)d_dμ

−qd(x₁).

(19)

Therefore, by (19), we obtain the following theorem.

Theorem 8 (Multiplication formula). FordÎN withd≡1 (mod 2), we have

˜

E(nα,q),w(h, 1|x) =

[d]n qα

[d]_−q

d−1

a=0

waqha(−1)qE˜_n(α_,q)d_,wd

h, 1|x+a d

.

5. Polynomials E˜(nα,q),w(h,k|x)andk=h In (3), we know that

˜

E(_nα_,q)_,w(h,k|x) =

Zp

· · ·

Zp

wx1+···+xk_[x

1+· · ·+xk+x]nqαq(h−1)x1+···+(h−k)xkdμ−q(x1)· · ·dμ−q(xk). Thus, we get

(qα−1)nE˜n(α,q),w(h,k|x) = [2]kq n

l=0

n l

(−1)n−l q

αlx

and

Therefore, by (3) and (20), we obtain the following theorem. Theorem 9. ForhÎ ℤ, aÎNandnÎℤ+, we have

Therefore, by (21), we obtain the following theorem. Theorem 10. FornÎ ℤ+, we have

Let E˜(nα,q),w(k,k|x) =E˜

Therefore, by (23), we obtain the following theorem. Theorem 12. FornÎ ℤ+, we have

From (15), we note that

wqkE˜(nα,q),w(k|x+ 1) +E˜

(α)

n,q,w(k|x) = [2]qE˜n(α,q),w(k−1|x). (25) It is easy to show that

(qα−1)nE˜(nα,q),w(k|0) =

By simple calculation, we get n

and

˜

E(nα,q),w(k|x) =

Zp

· · ·

Zp

wki=1xk_[x+

k

i=1

xk]nqαq

k

i=1(k−i)xi_dμ

−q(x1)· · ·dμ−q(xk)

=

n

l=0

n l

[x]n_qα−lqαlxE˜(_l,α_q)_,w(k|0)

=qxα_E˜(α)

q,w(k|0) + [x]qα n

,n∈Z+,

with the usual convention about replacing(E˜(q,αw)(k|0))nbyE˜

(α)

n,q,w(k|0). Putx= 0 in (25), we get

wqkE˜(nα,q),w(k|1) +E˜

(α)

n,q,w(k|0) = [2]qE˜n(α,q)(k−1|0). Thus, we have

wqk(qαE˜q(α,w)(k|0) + 1)n+E˜

(α)

n,q,w(k|0) = [2]qE˜n(α,q),w(k−1|0),

with the usual convention about replacing(E˜(qα,w)(k|0))nbyE˜(nα,q),w(k|0).

Acknowledgements

The authors express their gratitude to the referee for his/her valuable comments.

Authors’contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 19 September 2011 Accepted: 29 February 2012 Published: 29 February 2012

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